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REPORT  SAMSO- TR-75-297 
{SUPERSEDES  TR-73-220) 


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Titan  IIIC  Guidance  with  the  Carousel  VB 
Inertial  Guidance  System 


T.  E.  DARNAUD  and  J.B.SHAUL 


Guidance  and  Control  Division 
Engineering  Science  Operations 
The  Aerospace  Corporation 
El  Segundo,  Calif.  90245 


\ 

i 


15  July  1975 


Prepared  for 

SPACE  AND  MISSILE  SYSTEMS  ORGANIZATION 
AIR  FORCE  SYSTEMS  COMMAND 
Los  Angeles  Air  Force  Station 
P.0.  Box  92960,  Worldway  Postal  Center 
Los  Angeles,  Calif.  90009 


Approved 


1/ 

/J.  R.  Allder,  Director 
Guidance  and  Flight  Dynamics 
Subdivision 

Guidance  and  Control  Division 
Engineering  Science  Operations 


Cj).  

D7  Baxter,  Director 
Titan  III  Directorate 
Vehicle  Systems  Division 
Systems  Engineering  Operations 


Publication  of  this  report  does  not  constitute  Air  Force  approval  of 
the  report's  findings  or  conclusions.  It  is  published  only  for  the  exchange 
and  stimulation  of  ideas. 

^RICHARD  E.  WOLFSBERGER,  Cornel,  USAF 
Assistant  Program  Director 
Expendable  Launch  Vehicles  SPO 

This  report  supersedes  and  replaces  TR-0073(34 1 3 -02)- 1 , dated  29 
November  1972. 


UNCLASSIFIED 


SECURITY  CLASSIFICATION  OF  THIS  PAGE  fJWi*.  Data  Bntatad) 


REPORT  DOCUMENTATION  PAGE 


SA  MS0.¥TR  - 75-297 


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5.  TYPe  OP  REPORT  A PERIOD  COVEREO 


7.  AUTMORfsJ 


) T.  E.yfearnaudRMB  J.  RT^haul) 


PggTTW: 


ACT  OR  GRANT  NUMBERf.J 


9.  PERFORMING  ORGANIZATION  NAME  AND  ADDRESS 

The  Aerospace  Corporation 
El  Segundo,  Calif.  90245 


10.  PROGRAM  ELEMENT.  PROJECT.  TASK 
AREA  * WORK  UNIT  NUMBERS 


!S 


11.  CONTROLLING  OFFICE  NAME  AND  ADDRESS 

Space  and  Missile  Systems  Organization 
Los  Angeles  Air  Force  Station  — ■ 

Los  Angeles,  Calif.  90045  £ 


14.  MONITORING  AGENCY  NAME  A AODRESS (II  dt  Hat  ant  I mm  Control  I Inf  Ollleo)  IS.  SECURITY  CLASS,  (ol  Ihlo  tapoH) 

Unclassified 

IS*.  DECLASSIFICATION/DOWNGRADING 
SCHEDULE 


16.  distribution  statement  (oi  thio  Raport) 


Approved  for  public  release;  distribution  unlimited 


17.  DISTRIBUTION  STATEMENT  (ol  Iho  abotracl  antarad  In  Block  30,  II  dl  liar  an  I from  Ha  port) 


>9.  KEY  WOROS  ( Continue  on  nvirat  «id«  If  nKMitry  and  Identity  by  bloc*  nunbar) 


Guidance  Equations 
Inertial  Navigation  ~ 

Carousel  VB  -3 

, Rocket  Steering  Law 
\ Vehicle-Borne  Computer  Program 


BSTRACT  (Contlnua  on  ravmraa  alda  It  ttacaaaary  and  tdantlty  by  block  numb«0 

The  Titan  II^C  Standard  Space  Launch  Vehicle,  starting  with  Vehicle  26, 
will  use  the  Carousel  VB  Inertial  Guidance  System  for  navigation,  guidance, 
and  digital  flight  controls.  ''This  guidance  system  consists  of  a Carousel  VB 
Inertial  Measurement  Unit  (1MU)  and  a MAGIC  352  Missile  Guidance  Com- 
puter (MGC),  both  manufactured  by  Delco  Electronics,  General  Motors 
Corporation,  The  C-VB  IMU  is  a modification  of  the  C-IVB  inertial  navi- 
gator currently  in  airline  service  as  a primary  navigation  system. — ^ 


00  ruKM  1473 

irACSiMlLCl 


a 


UNCLASSIFIED 

SECURITY  CL  AMI  F 1C  AT  ION  OF  THIS  RAOC  !«*•«  DM*  «**•»•« 


SB®®®*** 


SECURITY  CLASSIFICATION  or  THIS  RA<3E(Trh«n  Dim  Bill) 


ABSTRACT  (Conttnumd) 


' A unique  feature  of  the  Carousel  IMU  is  that  two  of  the  three  gyro/ 
accelerometer  sets  are  revolved  at  1 rpm  with  respect  to  the  stable 
platform.  The  effect  is  to  partially  cancel  certain  instrument  errors 
associated  with  the  revolving  instruments.  Conversion  of  this  instrument 
from  commercial  airline  navigation  service  to  guidance  and  control  of 
the  rocket  boost  vehicle  presented  a guidance  software  design  task  that 
is  the  subject  of  this  paper. 


UNCLASSIFIED 

SECURITY  CLASSIFICATION  OF  TNI*  RAOCfOM*  Oat* 


PREFACE 


The  guidance  equations  derived  in  this  paper  were  developed  by 
the  authors  in  a joint  effort  with  D.  L.  Kleinbub  and  A.  C.  Liang  of  The 
Aerospace  Corporation. 

A summary  of  the  first  issue  of  this  report  was  presented  at  the 
Sixth  Hawaii  International  Conference  on  System  Sciences  January  11,  1973 
and  was  published  in  the  proceedings. 


CONTENTS 


m 


:\  m 


m 

fM 


:ni  poduction 


MISSION  SUMMARY  AND  VEHICLE  DESCRIPTION 


General 


Vehicle  Description  . . . . 

1 . Stage  0 

2.  Core  Stages  I and  II 


Rage  III 


Control  Module 


Mission  Description  (Flight  Plan  VIU 
1 . Ascent  to  Parking  Orbit 

2.  Parking  Orbit 

3.  Stage  HI  First  Burn 


4.  Transfer  Orbit 

5.  Final  Orbit  Injection 

6.  Paylcad  Separation  

GUIDANCE,  STEERING,  AND  NAVIGATION 


Gene  ral 


Guidance 


Thrust  Act  deration  Prediction 


Pitch  Stee  ring 
Yaw  Steering 


Estimate  of  Time  to  Go 


5.  Integral  Control  . . . . 

6.  Maneuvering  Equations 

Navigation 


Accelerometer  Resolution  and 
Compensation 


2.  Gyro  Drift  Compensation 


f PRECEDING  PIGS  BLANK-HOT  FUME] 

' T* 


-3- 


CONTENTS  (Continued) 


3.  Transformation  of  Velocity  to  Inertia] 

x,  y,  7,  Coordinate  System  

4.  Vehicle  Inertial  Position . 

5.  Gravity  Computations  

6.  Vehicle  Inertial  Velocity  

D.  Attitude  Errors  . 

1 . Synchros  

2.  Attitude  Error  Computations 

REFERENCES  


FIGURES 


1.  Titan  I1IC  External  Prolile  . ...  . 

2.  Typical  Titan  IIIC  Mission  Profile 

3.  Carousel  VB  IMU  Geometry 

4 Navigation  Block  Diagram 

TABLE 

1 . Carousel  VB  IGS  Characteristics  . - 


SECTION  I 


INTRODUCTION 


Referent  el  is  a complete  specification  of  the  guidance  equations 
for  the  Titan  HIC  launch  vehicle.  It  includes  all  information  necessary  for 
programming  the  launch  vehicle  guidance  computer.  Reference  1 does  not, 
however,  include  any  discussion  of  the  guidance  algorithms  or  the  derivation 
of  equations.  The  equations  specified  by  Ref.  1 and  discussed  in  this 
document  are  applicable  to  SSLV  C-26  and  subsequent  vehicles. 

Section  II,  a very  general  description  of  a typical  Flight  Plan  VII 
mission  and  of  the  Titan  IIIC  vehicle,  is  included  as  an  aid  to  understanding 
the  inflight  equations,  some  aspects  of  which  arc  vehicle-  and/or  rnission- 
peculia  r. 

Section  III  is  a discussion  of  the  Titan  IIIC  navigation,  guidance, 
and  steering. 


-5- 


SECTION  II 


MISSION  SUMMARY  AND  VEHICLE  DESCRIPTION 


A.  GENERAL 

The  Flight  Plan  VII  mission  consists  of  injecting  a payload(s)  into  a 
synchronous  or  near- synchronous  equatorial  orbit  using  the  Titan  IIIC  SSLV. 
The  combined  vehicle  Stages  II  and  III,  together  with  the  payload,  are 
injected  directly  into  an  e'liptical  parking  orbit  having  a perigee  of  approxi- 
mately 80  nmi  and  an  apogee  of  approximately  235  nmi.  At  the  first 
descending  or  ascending  node,  depending  on  mission  options,  a first  burn  of 
Stage  III  produces  an  elliptical  orbit  with  an  apogee  of  approximately  19,323 
nmi  and  an  orbital  inclination  . educed  by  approximately  2-1/4  deg.  At  apogee, 
a second  Stage  III  burn  produces  the  desired  synchronous  (or  near  synchronous) 
equatorial  orbit. 

B ■ VEHICLE  DESCRIPTION 

The  Titan  IIIC  launch  vehicle  shown  in  Fig.  1 consists  of  a solid 
motor  stage  and  three  liquid  engine  stages  plus  a control  module.  The  solid 
motor  stage  is  referred  to  as  Stage  0,  and  the  three  liquid  engine  stages  are 
referred  to  as  Core  Stages  I c II,  and  Stage  III.  The  control  module  and 
Stage  III  are  sometimes  collecti vely  called  the  transtage.  The  stages  are 
briefly^  described  in  the  following  paragraphs. 

1.  STAGE  0 

Stage  0 consists  of  two  solid  propellant  rocket  motors  positioned 
parallel  to  the  standard  core  in  the  yaw  plane.  Thrust  vector  control  is 
accomplished  by  injecting  oxidizer  (N^O^),  pressurized  by  gaseous  nitrogen 
(N^),  into  any  of  four  quadrants  of  the  nozzles. 


PRECuDINO  PAJS  BLANK-HOT  PIIirsB" " 


-7- 


1 


PAYLOAD  FAIRING- 


STATION 


FAIRING  AND  PAYLOAD- 
INTERFACE 

STAGE  il  AND  STAGE  III- 
SEPAR ATION  PLANE 


-CONTROL  MODULE 


STAGE  II 


STAGE  I A.'O  STAGE  II- 
SEPAR ATION  ’’LANE 


/'7°h 

' l\  ]/ — 24.  00  R 

PK\  “ ' - 360- 15 


| \ V-,z 


125.80 

(typ  each  segment) 


STAGE  I — 


-12.00  R 


STAGE  0 (2  SRMs)- 


5 SEGMENT  SOLID- 


6 OFFSET  v 


r T "1  14- 


-TVC  TANK 


H \ 


2°  OFF  SET  — 

S^O’  GIMBAL-Jf-^n  5° 3 O'  GIMBAL p 

(max) 


—1213.  10 
—1224. 3 1 
1240.68 
— 1274.  12 

1304.25 

—1326.00 

—1348.  10  (nozzle  exit) 
- 1375.79 


ROLL  AXIS 


HEATSHIELD  —r'/  b.L.O. 

THRUST  CHAMBER—'  | 
COVERS  ! 


-117°  24' 
(typ) 


85.00  R- 

(typ) 


<fc.  YAW  AXIS^jargex 


126. 51 

(typ)  I 


-W.  L.  60 


-<E  PITCH  AXIS 


Figure  1.  Titan  IIIC  External  Profile 


-8- 


z. 


CORE  STAGES  I AND  II 


Stages  I and  II  are  powered,  respectively,  by  dual  and  single  *hrust 
chamber,  turbopump-fed,  liquid  propulsion  systems.  These  systems  utilize 
storable  propellants,  a 50:50  mixture  of  hydrazine  and  unsymmetrical 
dimethylhydrazine  (UDMH)  for  fuel  and  nitrogen  tetroxide  (N^O^)  for  oxidizer. 

3.  STAGE  III 

The  Stage  III  propulsion  system  consists  of  two  pressure-fed  engines. 
Stage  III  also  contains  an  attitude  control  system  (ACS),  which  provides  atti- 
tude control  during  coast,  ullage  control  prior  to  main  engine  burns,  and 
velocity  additions  for  vernier  phases  and  for  satellite  ejections.  The  pro- 
pellants used  for  the  main  propulsion  system  are  the  same  as  for  Stages  I 
and  II.  The  attitude  control  system  employs  hydrazine  monopropellant 
engines. 

4.  CONTROL  MODULE 

The  control  module  contains  the  major  elements  of  the  flight  control, 
inertial  guidance,  electrical,  telemetry,  tracking,  and  flight  safety  systems. 
The  control  module  is  attached  to  the  forward  end  of  Stage  III  and  remains 
attached  throughout  flight. 

C.  MISSION  DESCRIPTION  (FLIGHT  PLAN  VII) 

The  following  is  a brief  mission  description  of  a typical  Flight  Plan 
VII.  All  values  given  are  only  approximate,  and  any  discussion  of  the  Flight 
Plan  that  does  not  contribute  to  understanding  the  guidance  equations  has  been 
intentionally  omitted.  A complete  description  of  a Flight  Plan  VII  mission 
is  given  in  Refs.  2 and  3.  Figure  2 is  an  illustration  of  the  Flight  Plan. 

1.  ASCENT  TO  PARKING  ORBIT 

The  vehicle  is  launched  vertically  from  Air  Force  Eastern  Test 
Range  (ETR)  Pad  41  or  40;  shortly  after  liftoff,  it  is  rolled  from  a pad  azimuth 
of  100.  2 deg  to  a flight  azimuth  of  93  deg.  Following  the  roll  maneuver,  an 
open-loop  pitchover,  with  load  relief  during  the  region  of  max  Q,  is  performed 
for  the  remainder  of  Stage  0 burn.  Stages  I and  II  are  closed-loop  guided; 


-9- 


ss 


•H 


the  engines  burn  for  approximately  149  and  207  sec,  respectively,  and  the 
stages  burn  to  propellant  exhaustion.  At  Stage  II  propellant  depletion,  the 
vehicle  is  nominally  in  an  elliptical  orbit  80  X 235  nmi.  Further,  in  the  case 
of  a low  performing  vehicle,  the  minimum  orbit  is  80  X 95  nmi. 

2.  PARKING  ORBIT 

Immediately  after  injection  into  the  parking  orbit,  the  vehicle  attitude 
is  adjusted  to  the  standard  orbital  orientation:  vehicle  longitudinal  axis 
normal  to  the  geocentric  radius  vector,  the  nose  of  the  vehicle  pointed  approxi- 
mately along  the  velocity  vector,  and  the  vehicle  pitch  plane  coincident  with 
the  orbital  plane. 

After  coasting  in  the  parking  orbit  to  near  the  f-rst  descending  or 
ascending  node,  the  vehicle  is  reoriented  in  yaw  as  a startup  attitude  for  the 
first  burn  of  the  transtage.  Concurrently,  a roll  maneuver  is  performed 
so  that  the  vehicle  telemetry  antenna  is  pointed  toward  the  desired  ground 
tracking  station  upon  completion  of  the  reorientation  maneuver. 

3.  STAGE  III  FIRST  BURN 

Close  to  the  equatorial  crossing,  the  Stage  III  engines  are  ignited 
and  burn  for  approximately  300  sec.  The  vehicle  is  injected  into  an  elliptical 
orbit  with  an  apogee  of  19,  323  nmi.  The  first  burn  includes  an  orbital  plane 
change  maneuver  of  approximately  2.  25  deg.  If  required,  the  ACS  of 
Stage  III  can  adjust  the  orbital  parameters  by  a vernier  velocity  addition. 
Nominally,  shutdown  of  the  Stage  III  engines  is  controlled  so  that  a 6- sec 
ACS  vernier  phase  is  required. 

4.  TRANSFER  ORBIT 

During  the  transfer  orbit,  the  transtage  performs  certain  maneuvers 
to  meet  thermal  control  requirements.  Currently,  three  options  axe  being 
considered  for  thermal  maneuver.  The  first,  called  rotisserie,  is  an  oscilla- 
tory roll  maneuver  of  approximately  ±115  deg  at  a rate  of  1 deg/sec,  with  a 


-11- 


f -min  dwell  time  at  cat.  h cxlremo  position.  In  the  sei  ond  maneuver, 
toasting.  th<  vehicle  is  simply  turned  hack  and  forth  in  space  at  widely- 
spaced  time  intervals,  l’he  third  is  a continuous  roll  maneuver  between  1 
and  2 deg/ set.  In  all  three  options,  the  relationship  of  the  sun  vector  to  the 
vehicle  or  spacecraft  axes  is  specified  by  the  payload  thermal  requirements. 
In  addition  to  the  thermal  maneuvers,  the  vehicle  is  oriented  several  times 
during  the  transfer  orbit  tc  an  attitude  that  permits  reliable  reception  of 
telemetry. 

Finally,  shortly  before  reignition  of  the  Stage  III  engines,  the 
vohic  le  is  reoriented  to  a startup  attitude  for  the  second  burn,  which  also 
points  the  telemetry  antenna  earthward. 

n.  FINAL  CRB  IT  INJECTION 

The  second  Stage  III  burn,  of  approximately  104- sec  duration, 
injects  the  payload  and  /ehicle  into  a circular  orbit  at  19,323-nmi  attitude, 
with  a near  zero-deg  inclination. 

6.  PAYLOAD  SEPARATION 

After  the  second  Stage  III  shutdown,  the  vehicle  is  reoriented  for 
the  satellite  separation  phase.  This  orientation  can  vary  depending  on  the 
selected  mission.  After  sufficient  time  for  stabilization  at  the  desired  atti- 
tude, the  ACS  is  switched  (o  the  payload  release  coast  mode;  after  sufficient 
time  for  stabilization  in  this  mode,  the  payloaa  is  released.  At  this  point, 
the  Titan  IIIC  mission  can  be  ended  with  a transtage  shutdown  sequence. 
Alternatively,  the  equations  provide  an  option  for  performing  a short 
propellant  settling  burn,  another  payload  release  sequence,  and,  finally,  the 
transtage  shutdown  sequence. 

In  both  of  these  payload  release  options,  the  equations  issue  a 
variety  of  required  discretes  and  can  perform  multiple  reorientations  as 
specified  by  payload  requirements. 


-12- 


SECTION  III 


GUIDANCE,  STEERING,  AND  NAVIGATION 


A.  GENERAL 

The  vehicleborne  digital  computer  program  is  divided  into  three 
distinct  sections:  digital  flight  control  equations,  ground  equations,  and 

guidance  equations.  This  document  treats  only  the  guidance  equations;  the 
other  programs  are  mentioned  only  when  they  interface  with  the  guidance 
equations.  The  carousel  sensor  and  gimbal  geometry  is  given  in  Fig.  3, 
while  principal  computer  and  sensor  characteristics  are  tabulated  in  Table  1. 
Figure  3 and  Table  1 are  modifications  of  data  taken  from  Ref.  9. 

The  guidance  equations  are  organized  into  various  subprograms. 

A main  control  or  executive  program  controls  logical  routing  to  appropriate 
calculations , depending  on  the  phase  of  flight  and/or  significant  flight  events. 
The  navigation  equations  accept  inertial  measurement  unit  (IMU)  data  and 
compute  current  vehicle  position  and  velocity.  Booster  steering  equations 
calculate  open-loop  steering  for  Stage  0.  Powered  flight  equations  provide 
vehicle  steering  as  a function  of  navigational  quantities  and  desired  end  con- 
ditions. Several  coast  phase  subprograms  provide  computations  for  desired 
vehicle  attitudes  and  event  initiations  during  nonpowered  flight.  Finally,  the 
specific  flight  plan  and  the  multistage  Titan  1IIC  vehicle  necessitate  numerous 
subprograms  to  repeatedly  reinitialize  the  guidance  equations  for  each  phase 
of  flight  and  to  issue  discretes  to  control  events  such  as  engine  shutdown, 
engine  ignition,  and  stage  separation.  The  following  paragraphs  describe 
the  guidance  philosophy  and  overall  logic  flow  of  the  guidance  equations. 

B.  GUIDANCE 

The  guidance  philosophy  governing  the  guidance  equations  presented 
in  this  document  is  commonly  termed  'explicit"  guidance.  Generally,  an 


-13- 


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explicit  guidance  scheme  is  one  in  which  an  explicit  solution  to  the  powered 
flight  dynamics  is  computed  in  flight  and  the  desired  end  result  of  guidam  e 
is  explicitly  defined  in  the  guidance  equations.  The  Flight  Plan  VII  guid- 
ance equations,  deoenclmg  upon  the  flight  phase,  define  different  desired 
end  conditions.  The  burns  of  Core  Stages  I and  II  are  planned  for  a specific 
position  magnitude,  velocity  magnitude,  radial  velocity,  and  preplanned 
orbit  planes.  The  end  result  of  guidance  for  the  first  burn  of  Stage  III 
(transtage)  is  injection  onto  a transfer  ellipse  whose  apogee,  semi-major 
axis  orientation,  and  orbital  plane  are  specified.  Finally,  during  the  second 
burn,  the  transtage  guides  to  a near-circular  orbit  at  the  transfer  ellipse 
apogee  radius  and  the  orbit  piano  is  specified  (approximately  equatorial). 

References  4,  5,  and  6 all  contain  general  discussions  of  explicit 
guidance  algorithms. 

1.  THRUST  ACCELERATION  PREDICTION 

A basic  requirement  of  the  guidance  equations  presented  here  is  a 
prediction  of  vehicle  performam  c,  particularly  of  thrust  acceleration.  For 
this  report,  "thrust  acceleration"  is  used  as  an  abbreviation  for  "accelera- 
tion (of  the  vehicle)  due  to  thrust.  " The  following  paragraphs  derive  a 
time-dependent  expression  for  thrust  acceleration. 

Consider  the  variable  T , defined  as 

r 

= W/W  (1) 

whe  re 

W - total  weight  of  vehicle  at  any  time 
W • rate  of  change  of  vehicle  weight 

T^  can  be  interpreted  as  the  time  remaining  to  zero  vehicle  weight  (mass 
intercept),  if  a constant  W is  assumed. 


-16- 


Since  specific  impulse  is 


1 = Thrust/  W 

sp 


and  thrust  acceleration  is 


Ath  --  Thrust/ (W/g) 
by  solving  for  thrust  and  weight  as 


(2) 


(3) 


W = T W 


Thrust  = W1 


sp 


(4) 


one  can  express  A„,,  as 
1 H 


A—u  - gl  /T 
TH  6 sp  r 


(5) 


Further,  since  the  effective  exhaust  velocity  is  given  as 


V = gl 
e 6 sp 


(6) 


then  it  follows  that 


ATH  = Ve/Tr 


(7) 


Finally,  since  T^  is  a linear  function  of  time,  then  A^,^  as  a 


function  of  time  t can  be  expressed  as 


ATH'1'  - Ve/(TrO*t) 


(8) 


where  TrQ  = value  of  Tr  computed  at  t = 0. 


-17- 


2.  PITCH  STEERING 

I"  polar  coordinates , the  two-body  equations  of  motion  for  a vehicle 
in  a central  for.  c field  gives  the  radial  acceleration  as: 


^ ^THr  ” 


wher< 


^THr  ~ rdciial  thrust  acceleration 


L.  n 

w *'■  centrifugal  acceleration 
2 

|i  / R ~ acceleration  due  to  gravity 

If  a time-dependent  guidance  steering  law  for  the  radial  thrust 
pointing  d1  rection  is  assumed  to  be 


ATHr/ATH  T A ' Bli  f WR2  - ci2R)/Ath] 
then,  upon  substituting  for  A^,  one  finds  that  Eq.  (9)  becomes 


RMA-tBt>ATH  (11) 

This  steering  law,  the  familiar  "linear  sine"  law,  has  been  applied 
to  numerous  guidance  programs.  It  has  the  advantages  of  simplicity  and 
very  near  fuel  optimality  for  current  ro''kof-r'-.'>'.vc'nc,i  u. 

Since  explicit  guidance  equations  require  a prediction  of  vehicle 
motion,  it  is  desirable  to  time-integrate  Eq.  (1  1}  twice  to  estimate  future 
vehicle  radial  velocity  and  position.  However,  before  these  integrations 
are  attempted,  Eq.  (11)  can  be  simplified  to  reduce  integration  complexities. 


- 18- 


The*  linear  assumption  of  T can  be  written  as 


t 


(12; 


Substituting  Eq.  (12)  into  Eq.  (11)  yields 


R = (A  + BTr0  - BTr)(ATH) 


(13: 


And,  after  some  algebraic  manipulation, 


R = -BT  Ath  + (A  + BTrQ)(ATH) 


= -BV  + (A  i BT  r) (A_u) 
e rO  TH 


(14) 


In  Eq.  (14),  the  quantities  A,B,Vo,  and  are  assumed  constant; 


therefore,  Eq.  (14)  can  be  rewritten  as 


R = A'  + B A 

r r T H 


(1  5> 


Time  integration  of  Eq.  (15)  from  a given  time  (t  = 0)  to  cutoff  (t  - T ) yields 


Vrr  V 


rT  rT 

+ / 8 A'  dt  + B / 8 A„udt 

r Jn  rJ  0TH 


(16) 


/•T  ft 

f,  % *> 


R,  = R + V T + 
f r g 


dt  + B*fn  8 f ATHdS 

JO  Jo 


dt 


-19- 


where 


&S*  ■'  s 

M 4 


WM 


|g|| 

il 


V = present  radial  velocity 
r 1 


V r - radial  velocity  at  cutoff 
rf 


iir 


%k4. 


m 


R = present  ra fiai  position 
= radial  position  at  cutoff 

The  integrals  of  thrust  acceleration  in  Eq.  (16)  can  be  evaluated 
using  the  time-dependent  expression  for  A^^,  given  in  Eq.  ( 1 Z) : 


nil 


h M 


r 

/ s 

Jo 


A.dt  - V 
Th  g 


A ds  di  - V T - V T 4 V T 
IH  g g g rO  eg 


in  these  integrals,  it  is  assumed  that  V is  known.  A derivation  of  the 
b g 

second  integral  (a^)  can  be  found  in  practically  any  discussion  of  an  explicit 
guidance  technique,  such  as  R ..  4. 

To  solve  for  the  steering  coefficients  A^  and  Bf>  one  must  substitute 
the  values  for  the  integrals  in  Eq.  (17)  into  Eos.  (16): 


V c - V 4 A'  T + B V 
rf  r r g r g 


R,  R + V T 4 A'  (T2/2)  4 B (a,  ,) 
I r g r g r 1 2 


it 


(19) 


Solving  for  yields 


B = R,  - R - (T  / 2)(  V , + V ) 
r f g ” rf  r 


Tro  - <y  2> 


+ V T 
e g 


and  for  A', 
r 


A'  = (1/T  )(V  , - V - V B ) 
r g rf  r g r 


(20) 


Finally,  the  expression  for  the  desired  radial  acceleration  A^  .s 
obtained  by  substitution  into  Eq.  (15): 


A 

r 


R = (1  /T  )(V  , - V 
g rf  r 


- V B ) + B A, 
g r'  r 


rH 


(21) 


3.  YAW  STEERING 

The  yaw  steering  is  derived  in  a manner  analogous  to  the  pitch 

steering  scheme.  In  the  yaw  case,  a desired  orbital  plane  is  specified  uy  a 

vector  normal  to  the  plane.  Normal  error  parameters  R and  can  be 

computed  as  the  dot  product  of  vehicle  inertial  position  and  velocity  with  the 

specified  normal  vector.  Since  the  desired  value  of  both  error  parameters 

is  zero  at  cutoff,  the  following  equations  can  be  used  to  determine  the  yaw 

steering  coefficients  A and  B : 
f n n 


0 = V + A T + B V 
n n g n g 


0 = R + V T + A T / 2 + B a,  _ 
n n g n g n 1 2 


(22) 


-21- 


4. 


ESTIMATE.  OF  TIME  TO  C.O 


* .1 


The  estimate  of  time  to  go  is  derived  from  AM,  the  vehicle  angular 
momentum  to  bo  gained,  which  is  given  as 


AM  = Mf  - M 


"M 


where 


r<; 


. « 


M^  = desired  vehicle  angular  momentum  magnitude  at  cutoff 
M - present  vehicle  angular  momentum  magnitude 


Angular  momentum  to  be  gained  is  also  given  by  the  integral  equation 


AM  = 


where  T ^ is  the  time  until  rocket  engine  cutoff. 

To  estimate  time  to  go,  solve  the  integral  Eq.  (24)  for  T by  a 

8 

numerical  integration  technique.  The  magnitude  of  angular  momentum  is 
given  as 


M = Rv 


v^  - velocity  in  the  tangential  direction 


R = position  magnitude 


Per  unit  mass 


Differentiating  Eq.  (25)  produces 


M = Rvt  + Rv^ 


(26) 


Another  expression  based  on  orbital  and  tangential  thrust  acceleration  is 


\ = -\R/R  + ATHt  (27) 

where  is  the  vehicle  thrust  acceleration  magnitude  in  the  tangential 

direction.  Substituting  the  expression  for  into  Eq.  (26)  yields  for  M the 
simple  expression 

M = Rvt  + R(-vtR/R  + ATHt)  = RATHt  (28) 


The  integral  of  Eq.  (24)  is  evaluated  using  Simpson's  integration 

formula,  with  the  integrand  computed  at  t = 0,  t = T , and  three  equally- 

6 

spaced  points  in  between. 


The  first  factor  in  the  integrand,  position  magnitude,  is  computed  at 
the  five  desired  time  points  by  a linear  interpolation  between  the  present 
vehicle  position  magnitude  R and  the  final  position  magnitude  R^;  thus, 


R(tj)  = R + (j  - l)(Rf  - R)/4  j = i,  ...5  (29) 

Thrust  acceleration  magnitude  as  a function  of  time  has  been  derived  m 
Paragraph  III.  B.  1.  Thus,  it  follows  that 


ath<V  ^ 


V 


J ' 


Tr0-  - 


J )T  / 4 
g 


j = 1. 


(30) 


Where  T is  tnc  time-to-go  from  the  last  major  cycle  decremented  by  1 sec 

g 


-23- 


The  tangential  acceleration  A,p^  is  obtained  from  a square  root  as 

, 1/2 


ATHt(y  = 


ATH(V  " ATHr{V  " ATHn(V 


(31) 


AIL  tliat  remains  is  to  compute  the  normal  and  radial  thrust  accelerations 

ATIIn  anc*  ATHr  at  Rie  re<5uire<^  <-irne  points. 

The  present  radial  components  of  thrust  acceleration  have  already 

been  given  as 


A 


THr 


a;  + BrATH  - (-p/R2  + u>2R) 


(32) 


and  at  cutoff  as 


A_ir  . - A'  + 
THrf  r 


B 


-ATHf  ” + wf Bf) 


(33) 


where  the  subscript  f designates  final  values  at  cutoff.  The  centrifugal 
acceleration  terms  in  Eqs.  (32)  and  (33)  can  be  rewritten,  respectively,  as 


2 

tjJ 


R = 


(34) 


? 2 
w*Rf  = M{ 


Finally,  by  linearly  interpolating  the  accelerations  due  to  gravity  and  centrifu- 
gal force  and  by  employing  the  value  of  acceleration  magnitude  derived  pre- 
viously in  Eq.  (30),  one  obtains  for  A^,^^  at  the  required  time  points 

ATHr(tj)  = Ar  + BrATH(tj}  " <^/r2  + m2/r3) 

- (j  - l){[(-p/R2  + M2/Rf3)  - (-p/R2  + M2/R3)]/4| 


(35) 


1,  ...  5 


-24- 


The  normal  component  of  thrust  acceleration  is  given  simply  as 


ATHn(V  = An  + BnATH(V  J = 1 > ■■■  5 (36) 

By  combining  the  results  of  Eqs.  (29)  and  (31),  one  obtains  the  integrand  of 
Eq.  (24)  at  the  specified  time  points 

M(t.)  = R(t.)ATHt(t.)  (37) 

and,  using  Simpson's  integration  rule,  one  can  compute  an  estimate  of  AM: 

AM'  = [M(t} ) + 4M(t2)  + 2M(t3)  + 4M(t4)  + M(t5)]T'/l2  (38) 

where  AM'  is  the  angular  momentum  if  the  initial  estimate  of  time  to  go  is 
assumed.  By  comparing  AM,  the  true  angular  momentum  to  be  gained,  with 
AM',  one  can  compute  an  adjustment  to  the  time  to  go  T 

it 

Ta  = (AM  - AM')/M(t5)  (39) 


Finally,  the  time  to  go  becomes 


T 

g 


= T'  + T 
g a 


(40) 


With  time  to  go  T and  the  time  to  mass  depletion  T , the  acceleration 

O 1* 

integral  V is  computed  as 
S 


V 

g 


Iv  /(T 
j e ' rO 


t)  dt  = -V  ln(l  - T IT  J 
e o'  r 0 


(41) 


-25- 


The  guidance  scheme  described  here  is  iterative.  The  steering  coefficient 
calculations  require  an  estimate  of  time  to  go  T , while  the  time-to-go 
calculations  presuppose  a knowledge  of  the  steering  coefficients.  Fortu- 
nately, experience  has  shown  that  with  reasonable  startup  values  and 
appropriate  updating  between  major  guidance  cycles,  the  equations  described 
here  ccnverge  quite  rapidly  for  the  mission  considered. 

5.  INTEGRAL  CONTROL 

In  Paragraphs  III.  B.  2 and  III.  B.  3,  expression's  were  derived  for 
desired  radial  and  normal  thrust  accelerations.  If  these  accelerations  were 
translated  directly  into  vehicle  axis  pointing-direction  commands,  a thrust 
acceleration  pointing  e**ror  would  result  because  of  vehicle  ard  IMU  prop- 
erties, such  as  engine  misalignments,  vehicle  center  of  gravity  effects, 
and  IMU  gimbal  misalignments.  Integral  control,  the  method  for  overcoming 
these  pointing  errors,  is  used  to  measure  differences  between  desired  and 
actual  guidance- computed  accelerations.  These  differences  are  numerically 
time-integrated  with  an  appropriate  weighting  factor  for  stability  reasons, 
i.e.  , a digital  filter,  and  finally  subtracted  from  the  desired  radial  and 
normal  accelerations.  The  final  steady  state  effect  is  to  bias  the  vehicle- 
pointing commands  so  that  the  observed  misalignments  are  canceled. 
Preference  7 contains  a complete  description  of  integral  control.  The 
equations  are  mechanized  as 


AA,rAAn  + (C:F6}  - AnJ  l4Z) 

AAr-AAr  + (cF6)  [(AVr/At)  - Aj  (43) 


The  arrow  indicates  that  the  results  are  stored  in  the  original  memory 
location . 


-26- 


where 


i 

i 

i 

> 


AA  and  AA 
n r 


integral  control  terms 


Cj.,^  integral  weighting  factor 
A^  - desired  normal  component  of  total  vehicle  acceleration 
A - desired  radial  component  of  total  vehicle  acceleration 


AV  /At  - guidance-computed  normal  component  of  total  vehicle 
acceleration 

AV  /At  guidance  - computed  radial  component  of  total  vehicle 
acceleration 

At  1 -second  major  cycle  interval 


The  corrected  desired  normal  and  radial  equations  are  then  given  as 


nc 


= A 


n 


kA 


(44) 


A 

rc 


2 

u>  r 


(45) 


Equation  (45)  includes  terms  for  gravity  and  centrifugal  accelerations,  these 
terms  were  added  so  that  A^,  excluding  the  effect  of  AAf)  would  represent 
the  desired  radial  thrust  acceleration. 

Dividing  Eqs.  (44)  and  (45)  by  thrust  acceleration  magnitude  yields 
the  desired  unit  vehicle  longitudinal  axis  in  the  radial,  normal,  and  tangential 
coordinates 


ID  = A /A_„ 
&n  nc  TH 

(46) 

ID  = A / A u 
&r  rc  TH 

(47) 

1/2 

U£t  * (*  - °*n  - °fr) 

(48) 

-27- 


Then  the  unit  desired  vehicle  longitudinal  roll  axis  vector  U|  is  converted 
to  inertial  coordinates  by  a matrix  multiplication  as 


fix" 

i 

C 

in> 

r-f- 

1 

- 

N 

U£n 

U- 

-U£r_ 

(49) 


wiiere  N is  the  matrix  that  relates  tangential,  normal,  and  radial  coordinates 
to  EC1  (x,  y, coordinates. 

The  desired  unit  vehicle  pitch  axis  is  arbitrary  from  a guidance 
standpoint  and  is  selected  by  the  guidance  equations  to  conform  to  vehicle 
and/or  telemetry  antenna  constraints.  Finally,  the  desired  unit  yaw  axis 
is  formed  from  U^  x to  complete  a right-hand  orthogonal  coordinate 
system. 

6.  MANEUVERING  EQUATIONS 


This  section  presents  a derivation  of  the  maneuvering  equations. 

In  this  paper,  maneuvering  is  considered  the  application  of  rate  limits  to 
the  guidance  commands  (desired  vehicle  axes)  and,  if  the  desired  maneuver 
is  large,  the  computation  of  an  efficient  maneuver.  In  addition,  the  maneuver- 
ing equations  calculate  values  to  interpolate  between  successive  major  cycle 
commands  on  a minor  cycle  basis.  The  attitude  error  equations,  performed 
on  a minor  cycle  basis,  take  the  output  commands  from  the  maneuvering 
equations  in  the  form  of  desired  vehicle  axis  vectors.  The  desired  vehicle 
axes  are  compared  with  the  actual  present  vehicle  axes  (read  from  the  1MU 
attitude  sensors),  and  vehicle  attitude  errors  are  computed.  The  flow  is 
from  guidance  equations  to  maneuvering  equations  to  attitude  error  equations. 


-28- 


The  initial  calculations  made  by  the  maneuvering  equations  are  a 
coordinate  conversion  from  earth  centered  inertial  (ECI)  to  drifted  launch- 
centered  inertial  (LCI),  as  follows: 

[rg]t  = [0]T  [*][cg] 

u£r  - [RG]  [U$] 

5 6 (50) 

U^  = [RG]  [Uc] 


where  U^c,  U^,  and  U^c  are  the  final  desired  commanded- vehicle  roll, 
pitch,  and  yaw  axes  in  drifted  LCI  coordinates:  [CG]  is  a coordinate  trans- 

formation matrix  from  ECI  to  LCI  coordinates;  [p]  is  the  present  compen- 
sable IMU  drift  matrix;  and  4*  is  a matrix  chosen  to  rearrange  the  rows  of  <p 
so  as  to  be  compatible  with  the  LCI  coordinate  system. 

The  drifted  LCI  coordinate  system^  is  used  throughout  the  following 
equations,  since  it  is  the  most  convenient  coordinate  system  for  the  guidance 
minor  cycle  calculations;  the  IMU  attitude  readouts  are  referenced  to  the 
drifted  LCI  coordinate  system.  The  drifted  LCI  system  (called  a,  b,  g)  can 
also  be  considered  an  ideal  gimbal  angle  system. 

Following  the  coordinate  transformation,  one  can  calculate  the 
transformation  matrix  of  the  desired  maneuver: 


u£c  ' 

u4a 

U!c- 

V 

u4c  ‘ 

u r 

M = 

^nc  ' 

°4a 

“V 

°na 

°nc- 

U„ 

4a 

_V ' 

uu 

V' 

V 

Uy 

4a 

where  Uc  , U , and  Ur  are  the  present  desired  commanded- vehicle  roll, 
pitch,  and  yaw  axes.  The  trace  of  the  BG  matrix  can  be  related  to  the 
commanded  maneuver  as 


-29- 


Trace  - 1 - 2 cos  8 


(52) 


o r 


cos  0 (1  - Trace)/2 

where  0 is  the  magnitude  of  the  maneuver. 

Cos  0 can  be  tested  to  determine  whether  to  command  a rate- 
limited  maneuver.  If  cos  0 is  sufficiently  close  to  1.0,  the  maneuver  is 
small  and  the  following  calculations  are  performed: 


dU!L  = U^c  ' UU 


and 


(53) 


ciU  T = U - U 

ol  0c  na 


The  variables  dU^  and  dU^j  represent  the  motion  of  the  commanded-body 
axes  Ut  and  U during  the  next  major  cycle.  If  dUf.  T and  dU  T are  multi- 
plied  by  the  reciprocal  of  the  number  of  minor  cycles  per  major  cycle,  C^q, 
the  commanded-body  axes  can  be  interpolated  on  a minor  cycle  basis.  With 


6R 

= dUt  T 

cm 

oP 

= dU  T 

cm 

nL 

(54) 


computed  on  a major  cycle  basis,  the  following  minor  cycle  calculations 
yield  the  interpolated  axes: 


-30- 


cm 


(55) 


U|dm 

^>ldm 


«-  u 


<-  u 


^dm 

qdm 


+ 6 R 
+ 6P 

cm 


If  cos  0 is  not  close  to  1.0,  a large  maneuver  has  been  commanded 
and  rate- limiting  is  desired.  The  maneuvering  strategy  selected  consists 
of  maneuvering  about  a single  inertially  fixed  axis  at  a constant  angular 
rate.  It  can  be  shown  (Euler's  theorem)  that  this  single-rotation  axis  is 
an  eigenvector  of  the  maneuver  transformation  matrix  given  in  Eq.  (51). 

The  following  paragraphs  describe  an  algorithm  for  computing  a command 
“eigenvector"  maneuver. 

To  solve  for  an  eigenvector,  E,  of  BG,  the  maneuver  transforma- 
tion matrix,  consider  a skew- symmetric  matrix  D,  calculated  as 

D = BG  - BGT  (56) 


The  D matrix  has  the  properties 


[D] 

[E] 

= to] 

(57) 

D. . 
!J 

= 0 

if  i = j 

(58) 

From  these  properties  of  the  D matrix,  the  following  set  of 
equations  is  derived: 


D12E2  + D13E3  " 0 


'D1 2E1  + D23E3  = 0 


-°13E1  ' D23E2  " 0 


(59) 


-31- 


Solving  for  the  eigenvector  in  Eqs.  (59)  yields 


E1  ' D23  BG23  ' BG32 

E2  ” "D1 3 " BG3 1 BG1  3 (60) 

E3  ^ D1  2 = BG1  2 ' BG21 

Alter  unitization,  the  eigenvector  is  transformed  from  body 
coordinates  to  the  LCI  coordinate  system  with  the  following  matrix 
multiplication: 


RL  = [A] 


UE. 


UE. 


UE 


3 J 


(61) 


where 


U 


UJ 


1 the  body  axes  as  columns. 


The  magnitude  of  the  eigenvector  computed  in  Eq.  (60)  is  propor- 
tional to  the  sine  of  the  maneuver  angle  0.  There  is  a singularity  wherein 
the  magnitude  of  the  eigenvector  approaches  zero  when  the  commanded 
maneuver  approaches  180  deg.  When  cos  0 is  sufficiently  close  to  -1.0, 
the  RL  vector  is  replaced  by  a unit- commanded  body  axis  according  to  the 
following  strategy. 

The  largest  diagonal  element  of  the  BG  matrix  is  determined. 


-32- 


Then,  if 


BG  largest,  R L.  = IL 
1 1 

BG,_  largest,  RL  _ U 

U L*  M < 


BG,,  largest,  lOl  = U 
33  t,a 


The  above  procedure  ensures  that  the  initial  maneuver  away  from 
the  180-deg  region  is  not  made  about  an  axis  which  is  perpendicular  to  the 
desired  maneuver  axis. 

RL  is  considered  the  unit  command  rotation  vector,  and  the 


variables  dU^^  and  dU^  are  computed  as 


dUf  j = (RL  X U^a)  MLIM 


dU  = (RLX  U ) MLIM 
PL  ' Pa/ 


and  the  interpolated  body  axes,  on  a minor  cycle  basis,  are  computed  as 
described  in  Eqs.  (54)  and  (55).  MLIM  is  the  magnitude  of  the  rate  limit. 

The  method  used  to  interpolate  the  commanded-body  vector  throughout 
the  major  cycle  results  in  a very  small  nonorthonormality  in  the  commanded 
vehicle  axes.  To  prevent  this  nonorthonormality  from  growing,  one  can  per- 
form an  orthonormalizing  process  for  each  major  cycle,  as  follows: 

Lu  = + dTJei.)/llJu + dLiJ 

L,a=1haX  %a*«OnL,/|U5aX  (U,a  + dUnL)l  (64) 

«na  = %X  V 


This  process  resets  the  command- vehicle  coordinates  to  orthonormality 
every  major  cycle. 


-33- 


c. 


NAVIGATION 


The-  guidance  navigation  calculations  compute  the  'present  vehicle 
position  and  velocity  using  as  inputs  accumulated  incremental  velocity  pulses 
from  a triad  of  force  rebalance  integrating  accelerometers.  One  of  the 
accelerometers  (WC1  is  mounted  on  the  IV U stable  platform  (turret);  the 
other  two  (l;C  and  VC)  are  mounted  on  a platform  that  rotates,  or  carousels, 
at  I rpm.  Figure  4 >s  a block  diagram  of  the  major  and  minor  cycle 
navigation . 

1.  ACCEL  fill  O V 1 1 T E R RESOLUTION  AND  COMPENSATION 


For  the  navigation  function  to  he  performed,  the  initial  azimuth 
(at  "go  inertial")  of  the  rotating  platform  must  be  determined.  Further, 
whenever  the  incremental  velocities  are  sampled,  this  azimuth  must  be 
updated. 

Four  coordinate  systems  are  used  in  the  accelerometer  resolution 
and  compensation: 


[uc,  vc,  wcj 


[l,2,3] 


[u,  v,w] 


[x,y, z] 


Actual  ca rouselling  accelerometer  input  axes 
coordinate  system.  The  coordinate  system  is,  in 
general,  nonorthogonal  and  rotating  due  to  1MU  drift 
and  ca  rouselling. 

Ideal  carouselling  coordinate  system  This  coordi_- 
nate  system  of  convenience  is  defined  as  follows.  3 
is  colinear  with  the  carouselling  axis  of  rotation. 

2 is  perpendicular  to  3;  it  is  in_the  plane  of  3 and  the 
vc  accelerometer  input  axis.  1 completes  the 
orthogonal  right-hand  set. 

Drifted  launch  site  coordinate  system.  Initially 
equal  to  the  initial  vehicle  roll,  pitch,  yaw  coordi- 
nate system.  After  go  inertial,  related  to  the  initial 
vehicle  coordinate  system  by  the  IMU  drift  matrix. 

Earth  centered  inertial  coordinate  system.  Defined 
as  follows: 

x = unit  vector  in  the  equatorial  and  prime  meridian 
planes  at  go-inertial  time. 

z = unit  earth  spin  vector,  i.e.,  North  Pole 

y - z x x to  complete  a right-hand  set. 


34 


Figure  4.  Navigation  Block  Diagram 


An  miii.il  >1/. 1 ‘nuih,  or  phase  angle,  is  computed  in  the  ground-in- 
flight interface  program  as  follows: 


FANG  - HTHYN  + CFANG  + CZZANG 


(65) 


where 


HTHYN  = the  angle,  computed  in  the  ground  alignment  program,  from 
north  to  the  VC  accelerometer  input  axis  plus  gyro  mis- 
alignment errors 

CZZANG  = gyro  misalignment  IMU  compensation  parameter 

CFANG  = parameter  to  convert  the  phase  angle  from  northeast  to 

dov/nrange-crossrange  coordinates  and  to  time  synchronize 
with  the  accelerometer  readouts. 

The  sine  and  cosine  of  the  initial  phase  angle  are  then  taken  for  resolution 
purposes . 


SS  = sin(FANG) 
BS  = cos  (FANG) 


(66) 


After  initialization,  the  carousel  phase  angle  sine  and  cosine  is 
updated  every  40  msec  by  a trigonometric  identity  as  follows: 


SSn+l  = <SSnHcos  0.24°)  + (BS  )(sin  0.24°) 
BSn+1  = (BSn)(cos  0.24°)  - (SSJisin  0.24°) 


(67) 


where  0.24  deg  is  the  angle  :’car  luselled"  through  in  40  msec.  Also,  the 
carousel  phase  angle  is  reinitiaJ  zed  to  SS  and  BS  at  1 -minute  intervals. 

The  accelerometer  incremental  counts  are  modified  with  the  calibra- 
tion constants  determined  as  part  of  factory  calibration  and  pad  test  procedure 
The  40-msec  frequency  accelerometer  compensation  equations  are 


AVuc  = (CKAX)(AN  - CKBXG)  + (CKNX)(AN  )2 

uc 

A ^vc  (CKAY)(ANvc  - CKBYG)  + (CKNY)(ANv(J2 

AVwc  = (HZAF)(AN  - HBZ)  + (CKNZ)(AN  )2 

wc 

whe  re 


ANuc’  ANvc’  ANwc  the  raw  counts  from  the  carouselling  uc  and  vc 

and  the  stationary  wc  accelerometers  over  the 
last  40-msec  interval 

CKBXG,  CKBYG  = the  uc  and  vc  accelerometer  factory  bias 

calibration 


HBZ  - ^e  wc  accelerometer  bias  calibration  computed 
during  final  align 

CKAX,  CKAY  = the  uc  and  vc  factory  scale  factor  calibration 

HZAF  = the  wc  accelerometer  scale  factor  calibration 
computed  during  final  align 


CKNX,  CKNY,  CKNZ  = the  uc,  vc, 


and  wc  nonlinearity  calibration 


Continuing,  one  finds  that 


ST1  = AVuc  " (czzb2U)(AV  ) - (CZZB3U)(A V t 

vc  wc’ 

ST2  = AV  - (CZZB3V)(AV  ) 

VC  WC' 


(69) 


Equation  (69)  shows  small  angle  approximations  to  misalignment 
rotation  compensation  where 


ST1 , ST2  = velocity  increments  in  1,1,3  coordinates 

CZZB2U  - misalignment  of  the  u axis  toward  ~Z 

CZZB3U  = misalignment  of  the  u axis  toward  3 

CZZB3V  = misalignment  of  tne  v axis  toward  3 


-37- 


ST1  and  ST2  are  then  resolved  into  u,v,w  coordinates  as  follows: 


IBl 


. 


mm 


AVum  = (BSHST1)  - (SS)(ST2) 


AVvm  = (SS)(ST1)  + (BS)(ST2) 


The  resolved  minor  cycle  velocity  increments  are  finally 
accumulated  as  follows: 


A V <-  AV  A V 
up  up  um 


AV  AV  + AV 
vp  vp  vm 


[T 


J I 

■ *>W'!  * 


A V f AV  + A V 
wap  wap  wc 


Once  per  second,  the  accumulated  velocity  increments  are  sampled 
for  use  as  inputs  to  the  major  cycle  navigation  equations 


AV 

«-  AV 

u 

up 

AV 

*-  AV 

V 

vp 

AV 

«-  AV 

wa 

wap 

and  AV 

are  chei 

up  vp  wap 

further  accumulation  in  the  next  major  cycle  interval. 


AV  0 
up 


AV  «-  0 
vp 


AV  <-  C 
wap 


The  w-accelerometer  misalignments  are  performed  on  a major 
cycle  basis,  since  the  w instruments  do  not  carousel.  These  misalignments 
do  rotate,  because  the  w-accelerometer  is  rotating  at  earth  rate  at  liftoff. 
During  guidance  initialization,  misalignments  are  computed  in  the  u,  v,  w 
coordinate  system  as  a function  of  the  initial  gimbal  angle 


bWl  = (CKB4W)(sin  aj  + (CKB5W)(cos  a) 


BW2  = - (CKB4W)(cos  a.)  + (CKB5W)(sin  a.) 


where 


BWl  - w-accelerometer  misalignment  away  from  u axis 

BW2  - w-accelercmeter  misalignment  away  from  v axis 

CKB4W,  CKB5W  = factory-calibrated  w-accelerometer  misalignments 

in  an  a gimbal-oriented  coordinate  system 


The  w-accelerometer  input  axis  also  "cones"  after  go  inertial  as  a 
function  of  the  inner  gimbal  angle,  because  of  misalignment  of  the  inner 
gimbal  axis  and  the  carouselling  axis.  The  complete  equation  for 
w-accelerometer  misalignment  compensation  is  given  in  Eq.  (75): 


AV  = A V + [BWl  + (CKBSW)(cos  a)  - (CKBCW)(sin  a)]AV 
w w & u. 

+ [BW2  - (CKBCWMcos  a)  - (CKBSW)(sin  »)]AV 


CKBSW,  CKBCW  = calibrated  carouselling  axis  misalignments  in  an 

a gimbal  angle  coordinate  system 


-39- 


2. 


GYRO  DRIFT  COMPENSATION 

Gyro  drift  compensation  is  accomplished  by  computing  the  platform 
drift  rate  with  calibration  values  obtained  during  hangar  and  pad  tests.  Then 
by  time -integration  of  the  drift  rate,  a matrix  p can  be  computed  that  trans- 
forms the  present  (u,v,w)  axes  to  the  initial  nondrifting  axes.  Further 
matrix  operations  then  transform  to  the  desired  ECI  coordinate  system. 


rAV  l 

rAV  -i 

sx 

u 

AV 

sy 

= [CG]r[4>]T[p] 

AV 

V 

AV 

L szJ 

AV 

L w_ 

(76) 


where 


AV 

sx, sy, sz 


measured  (sensed)  values  of  velocity  increments  in  the 
ECI  coordinate  system 


AV 

u,  v,  w 


measured  value  of  velocity  increments  along  the,  present 
drifted  launch  site  coordinate  system 


p - d rift  matrix 

4*  row  rearrangement  matrix 

CG  - coordinate  transformation  matrix  from  LCI  to  ECI 
coordinates 


Note  that  p,  cp  and  CG  are  shared  by  guidance  (Paragraph  III.B.6)  and 
navigation. 

If  the  instantaneous  drift  rate  vector  is  given  as  (pu>  pv,  pw),  the 
derivative  of  each  column  of  the  p matrix  is  the  cross-product  of  the  drift 
rate  vector  with  that  column  vector.  Calculating  the  cross  products  and 
combining  terms,  one  obtains  the  matrix  form 


p = [p] 


0 

6 

\ 


w 


-p 

0 

< P , 


w 


-p 


u 


0 


(77) 


Integration  of  p is  accomplished  by  using  a second  order  Runge- 
Kutta  algorithm.  First,  the  p matrix  is  integrated  a half  cycle  with  an 
initial  derivative: 


K~~  *n  + <At/2)*n  (78) 

The  drift  rate  is  then  averaged  over  two  cycles: 

A p = (A  p + A p , , ) / 2 
ru  run  run+l 

A p = (A  p + Ap  , . ) / 2 ( < ^) 

rv  rvn  rvn+l' 

A?  = (A p + A 0 , , )/2 

Tw  rwn  rwn+l' 


A final  step  advances  p as  follows: 


n+1 


1 Pn  + 


[p  1 
L Vn  , 


0 

A ^ 

-Ap 

rw 

Ap 

rw 

0 

-A?u 

-Ap 

rv 

Ap 
y u 

0 

(80) 


Computation  of  the  drift  vector  ip  in  u,  v,w  coordinates  is  complicated 
by  the  rotation  of  the  carouselling  instruments.  Drifts  of  each  gyro  must  be 
located  spatially  due  to  the  arbitrary  location  of  the  1MU  turret  and  the 
rotation  of  the  platform  at  1 rpm. 

All  drift  matrix  calculations  are  performed  once  per  second.  A 
central  assumption  for  the  drift  calculations  is  that  the  drift  occurred  approxi- 
mately centered  in  the  1 sec  compute  cycle.  To  locate  the  carouselling 
instruments,  one  must  compute  backed-up  values  of  the  variables  BS  and  SS. 


SSB  = -(BS)(sin  3°)  + (SS)(cos  3°) 


BSD  - (BS)(cos  3°)  + (SS){sm  3°) 


(81) 


The  variables  SSB  and  BSB  are  then  used  to  resolve  AV  and  AV  into 

u v 

coordinates  representing  the  "average"  position  of  the  carouselling  gyros 
over  the  last  second. 


AVgi  = (AVu)(BSB)  + (AVv)(SSB) 


AVg2  = -(A~Vu)(SSB)  + (AVv)(BSB) 


(82) 


The  fixed-torque  and  unbalance  drifts  of  the  carouselling  gyros  are  then 
computed  as 

DRFTU  FT  DU  + (CKU1)(AV  t)  + (CKU2)(AVg2) 

DRFTV  = FTDV  + (CKU3)(AVg2)  + (CKU4)(AVgJ) 


(83) 


where 

FTDU,  V = fixed-torque  drifts 
CKU 1 , 3 = spin-axis  unbalance  drifts 
CKU2,  4 = input-axis  unbalance  drifts 


In  addition  to  the  usual  error  sources  associated  with  an  IMU, 
carouselling  itself  introduces  gyro  drifts.  The  compensable  ones  are  called 
gimbal- oriented  bias  (GOB),  turret-oriented  bias  (TOB),  and  turret-oriented 
eta  (TOE).  GOB  is  a drift  in  the  carouselling  plane,  which  rotates  as  the  inner 
gimbal  rotates;  i.  e.  , a drift  fixed  to  the  inner  gimbal.  Initially,  GOB  is 
computed  in  launch-centered  coordinates  as 


-42- 


GOBC1  = (CK20)(cos  CK21) 
GCBC2  = (CK20)(sin  CK2.1) 


(84) 


where 


CK20  - GOB  magnitude 
CK21  = GOB  phase  angle 

7 

Later,  in  flight,  the  computation  of  GOB  reflects  the  rotation  about 
the  inner  gimbal  as  follows: 


GOBI  - - (GOB  C2)(cos  a)  + (GOB  Cl)(sin  a) 
GOB2  = (GOB  Cl  )(cos  a)  + (GOB  C2)(sin  a) 


(85) 


I urret-  or  tented  bias  is  a drift  of  the  tui  ret  (perpendicular  to  the 
carouselling  plane)  that  is  a function  of  several  harmonics  of  the  turret  present 
position  with  respect  to  the  inner  gimbal.  To  locate  the  turret  during 
guidance  initialization,  one  must  compute  the  angle  of  the  turret  revolution 
from  a zero  inner  gimbal  angle  value: 

- tan  (sin  Q-. /cos  a. ) 

where 


°'1  - initial  (at  go  inertial)  inner  gimbal  angle 
sin  a.  = sine  of  initial  a 
cos  or.  = cosine  of  initial  a 


-43- 


Then,  in  flight,  this  a,ngle  is  updated  as 


- tan  (sin  a! cos  a)  + 


(87) 


The  equation  for  TOB  is 


4 

TOB  = [cos  (n  a - 0 ) 

f— { iL  'it  l 

i=i 


(88) 


In  Eq.  (88),  n.  represents  integers  specifying  the  harmonics  of  a ^ to 
be  included  in  the  TOB  calculation,,  6^  represents  the  phase  angle  of  each 
harmonic;  and  K represents  magnitudes. 

TOE  is  a drift  in  the  carouselling  plane  that  is  a function  of  the 
location  of  the  turret  with  respect  to  the  launch- centered  coordinate  system. 
It  is  computed  at  guidance  initialization  as 


TOE  1 = - (CK22)[  (sin  a.)(cos  CK23)  + (cos  *.)(sin  CK23)  J 
TOE2  = (CK22)l(cos  m)(cos  CK23)  - (sin  a.)(sin  CK23)  ] 


(89) 


who  re 


CK22  TOE  bias  drift  magnitude 
CK23  TOE  bias  drift  phase  angle 

Finally,  the  derivative  of  <p  is  computed  as  shown  in  Eq.  (90): 


A pu  = (BSBKDRFTU)  - (SSB)(DRFTV)  + GOBI  + TOE1 
Apv  = (SSBMDRFTU)  + (BSB)(DRFTV)  + GOB2  + TOE2  (90) 

A pw  = HRZ  + (DIAU)(AV  ) - (CZZU6)[  (AVv)(sin  a.)  + (AVJfcos  ».)]  + TOB 


-44- 


V 


whe  "e 


3. 


BSB,  SSB  = sine  and  cosine  of  the  average  carousel  angle 
during  last  compute  cycle 

DRFTU,  DRFTV  = fixed-torque  and  g-sensitive  drifts  of  the  UC  and 

VC  carouselling  gyros 

GOBI,  GOB 2 = GOB  drift  [Eq.  (84)] 

TOE  1 , TOE 2 = TOE  drift  [Eq.  (89)] 

TOB  = TOB  drift  [Eq.  (88)] 

HRZ  = noncarouselling  Z gyro  fixed-torque  drift 

DIAU  - spin-axis  unbalance  drift 

CZZU6  = input-axis  unbalance  drift,  Z gyro 

TRANSFORMATION  OF  VELOCITY  TO  INERTIAL  x,y,z 
COORDINATE  SYSTEM 


The  inertial  x,  y,z  coordinate  system  is  defined  as  follows: 

x = unit  vector  in  the  equatorial  and  prime  meridian  planes  at 
go  inertial  time 

z = unit  earth  spin  vector,  i.  e.  , North  Pole 
y = z X x to  complete  a right-hand  set 


4.  VEHICLE  INERTIAL  POSITION 

The  following  trapezoidal  integration  formulas  are  used  to  obtain 
the  vehicle  position  (X,  Y,  Z)  in  the  inertial  coordinate  system: 

X <■  X + V (At)  + (1/2)  (At)2  (AV  + AV  ) 

Y «-  Y + V (At)  + (1/2)  (At)2  (AV  + AV  ) (9D 

y y o y 

Z Z + V (At)  + (1/2)  (At)2  (AV  + AV  ) 

Z b Z gZ 


-45. 


where 


V , V , V 
x y z 


At 


AV  , A V 
sx 

AV  , A V 

gx 


sy 


gy 


» A V 


sz 


,AV 


gz 


x,  y,  z components  of  vehicle  velocity 


major  compute  cycle  interval  - 1 second 


x,y,z  components  of  sensed  vehicle  velocity 
increment  from  Eq.  (76) 

x,  y,  z components  of  velocity  increments  due  to 
gravity 


Vehicle  position  magnitude  is  then  calculated  as 

2 2 2 1/2 

r = (x  + y + z ) 


(92) 


5.  GRAVITY  COMPUTATIONS 

An  approximation  for  the  gravitational  potential  of  the  earth  is 

U 

U = (n/R)[l  + (Ja2/3R2)(1  - 3 sin2\)  ] (93) 

where 


p = gravitational  parameter  of  the  earth 
R = distance  from  center  of  the  earth 
\ - latitude 

a = equatorial  radius  of  the  earth 

J = a characteristic  constant  that  is  a function  of  the  moments  of 
inertia  with  respect  to  the  polar  and  equatorial  axes 

A derivation  of  Eq.  (93)  is  contained  in  Ref.  7.  This  approximation, 
which  is  a truncated  series,  contributes  negligible  inaccuracies  to  the  overall 
navigation  function. 


-46- 


Then,  from  the  definition  of  the  Z component  of  vehicle  position, 
it  follows  that 

sin2  \ = 22/R2  (94) 

which,  upon  substitution  into  Eq.  (93),  yields 

U = (p/R)  (1  + Ja?'/3R2  - Ja2Z2/R4)  (95) 

The  x,  y,  z components  of  the  acceleration  due  to  gravity  are  then 
obtained  by  the  partial  differentiation  of  Eq.  (95)  with  respect  to  each  axis, 
and 

gx  = -aU/9x  = -(8U/9R)OR/ax) 

gy  = -au/ay  = (-au/aR)OR/ay)  (96) 

g = -au/az  = (-au/aR)OR/az) 

The  negative  sign  is  added  by  convention. 

An  evaluation  of  the  common  term  3U/aR  yields 

aU/aR  = (-p/R2)(l  + Ja2/R2  - 5Ja2Z2/R4)  (97) 

2 

and,  by  factoring  out  - -(p/R  ),  one  obtains 

aU/3R  = A (1  - Ja2A  /p  + 5 Jr.2 A Z2/pR2)  (98) 

8 8 8 


-47- 


In  addition,  one  finds  that 


dR/dx  = X/R 
9R/9y  = Y/R 
dR/dx  = Z/R 


(99) 


and 


aU/az  = ZfxJai 2Z/R5  = +2JaZA2Z/|j.R 

8 


(100) 


The  velocity  increments  over  a major  compute  cycle  due  to  gravi- 
tational acceleration  arc  given  in  Eq.  (101). 


AVgx  gx  -(ou/aR)(aR/ax) 


a v g - -(au/aR)(aR/ax) 

Bi  y 


AV  =g  - -OU/9R)OR/8z) 

gz  bz 


6.  VEHICLE  INERTIAL  VELOCITY 

The  inertial  velocity  of  the  vehicle  is  computed  as 


(101) 


(V 


.).  (V)  + (1  /2)[(AV  .)  , + (AV  .)  ] [At]  + (AV  )(At) 

in)  1 in  L gi  n+1  gi  nJ  1 J si 


i = x,y,  z 


gi 


(102) 


Averaging  AV  over  two  cycles  produces  a trapezoidal  integration  of  acceler- 


D.  ATTITUDE  ERRORS 

Attitude  errors  art?  < rucial  outputs  of  the  guidance  equations  utilized 
by  the  Digital  Flight  Controls  Equation  in  stabilizing  the  vehicle  to  a desired 
attitude.  They  represent  differences  between  the  present  actual  vehicle 
attitude  and  the  present  desired  vehicle  attitude.  The  desired  attitude  is 
computed  by  the  guidance  maneuvering  equations  (Paragraph  III.  B.  6);  the 
actual  attitude  is  derived  from  synchros  attached  to  each  gimbal. 


1 . 


form 


SYNCHROS 

The  synchro  signals  supplied  to  the  inflight  computer  are  of  the 


VI  = K sin  (0-120°)  and  V2  = K sin  (0-60°) 


(103) 


where  G can  be  any  of  three  gimbal  angles,  a,  p,  or  y„,  and  K is  a scaling 
factor  common  to  VI  and  V2. 

By  trigonometric  identities,  one  obtains 

VI  - V2  = K sin  (0-120°)  - K sin  (8-60°) 


= K[(sin  0 cos  120°  - cos  0 sin  120°) 
- (sin  6 cos  60°  - cos  0 sin  60°)] 

= K sin  0 (cos  120°  - cos  60°) 


(104) 


-K  sin  8 


-49- 


a n<l 

VI  I V2  K sin  (0  - 120°)  + K sin  (0  - 60°) 

K | (sin  0 cos  120°  - cos  0 sin  120°) 

+ (sin  0 cos  60°  - cos  0 sin  60°)  | 

(105) 

- K cos  0 (-  sin  120°  - sin  60°) 

= -K*/3  cos  0 

Thus,  by  simple  sum  and  difference,  one  obtains  the  sine  and 
cosine  of  the  gimbal  angles,  except  for  a common  multiplier  -K  and  the 
constant  V 3. 

Since  the  sine  and  cosine  of  the  gimbal  angles  are  desired  on  a 
minor  cycle  basis  as  a convenience  for  the  computation  of  vehicle  attitude 
errors,  it  is  necessary  to  determine  the  scaling  factor  K. 

At  go  inertial,  the  following  initial  calibration  is  performed  for 
each  of  the  three  gimbal  angles: 

X - V2  - VI 
Y = (VI  + V2)(C1) 

(106) 

K2  - (X)(X)  + ( Y)( Y) 

K -v/  K2 

In  this  series  of  equations.  Cl  is  a constant  equal  to  -1.0/\f3,  and  K 
is  the  calibrated  value  of  the  synchro  scale  factor. 

On  succeeding  minor  cycles,  the  sine  and  cosine  of  each  gimbal 
angle  are  computed  as  follows: 


-50- 


X = V2  - VI 
m 

Ym  = <V1  + V2>(C1) 

K2  - (X  )(X  ) + (Y  )(Y  ) 

m m m m m 

K - (1/2)  fK2  /K  + K ) 
m m m rn 


107) 


sin  0 = X/K 
cos  0 = Y/K 


m 


m 


These  minor  cycle  computations  use  Newton’s  square-root  algorithms, 
which  converge  quite  rapidly  if  the  initial  value  is  relatively  accurate. 

Since  the  three  gimbals,  n,  (3,  and  y^,  are  not  necessarily  zero  at 
go  inertial,  they  are  read  initially;  the  initial  values  are  used  to  compute  the 
three  angles  through  which  the  vehicle  rotates  during  the  flight.  Taking  a as 
an  example,  trigonometric  identities  yield 


sin  a = sm  a cos  a.  - cos  a sin  a 
pi  pi 


cos  a = cos  a cos  a.  + sin  a sin  a 
pi  pi 


(108) 


where 


sin  a,  cos  a = sine  and  cosine  of  vehicle  angular  rotation 
sin  a , cos  a = present  gimbal  synchro  sine  and  cosine  readings 
sin  o , cos  m = initial  gimbal  synchro  sine  and  cosine  readings 


An  identical  procedure  is  followed  to  compute  the  sine  and  cosine  of  the  }3 
and  vD  vehicle  rotations. 


-51- 


The  fourth  gimbal  y has  very  limited  travel  (‘"10  deg),  and  the 
following  simplifying  approximations  are  made: 

cos y - 1.0 


(109) 


sin  y ~ (CKDSCA}(sin  'y  ) - sin  y^ 


where  CKDSCA  = calibrated  scale  factor  for  sin 


ATTITUDE  ERROR  COMPUTATIONS 


The  IMU  gimbal  angles  (a,  p,  Y and  Y^)  ideally  establish  the 
relationship  between  two  vehicle  coordinate  systems:  the  present  roll, 
pitch,  and  yaw  (£,q,p  body  axes;  and  the  initial  body  axes  (£.,C,£.l.  Initial 
alignment  is  such  that,  at  launch,  a positive  roll  results  in  a negative  gimbal 
angle  «;  a positive  yaw  results  in  a positive  gimbal  angle  (3;  and  a positive 
pit<  h results  in  a positive  gimbal  angle  Y^.  The  fourth  gimbal  Y,  in  normal 
circumstances,  is  zeroed.  Its  sign  convention  is  the  same  as  Y 

i\. 


[uc.r,|.u, 


CO*  tp  0 -sin 

0 I 0 

\K  <»  * o«  \ 


('  sin  P o i » ■<*>  > 

i 

- sin  ,1  ( ,1  1(1  (j 

I I 


ljr,  0 


.]  (110) 


ij  si n it  i 06 


I 06  / | 


Performing  the  matrix  multiplication  with  cos  y ~ 1.0  yields 


-52- 





('  '>■>  1 1(,  ‘ " i“i  ) l(  --'ii  'i ) 

(»  ■ >&  \ ^ sin  6 cos  u 

( - cos  ^ ^ sin  >i  sin  / 

- cos  cos  0 sin  \ sin  a 

- cos  y^  c ob  3 sm  \ cos  a 

- -nn  ^ sin  o) 

- sin  Yp  cos  <>) 

(-  bin  |i) 

(t  os  3 cos  o + 

( - cos  (1  sin  a + sin  3 sin  \ tos 

sir*  3 sm  y s in  a 

(sin  > ^ cos  il  t cos  y ^ sin  y 

(sin  Y{^  sin  3 cos  a 

(-  sin  y ^ sin  3 sin  u 

- sin  cos  p sin  y sin  o 

- 6in  y^  cos  3 sin  y cos  a 

+ cos  sin  ») 

f cos  y . Cos  o) 

:n  i) 


or  M = lUg,  U^,  U^j 


If  the  matrix  of  initial  vehicle  axes  is  assumed  to  be  identity,  the 
rows  of  the  M matrix  comprise  the  present  measured  vehicle  axes  in  a 
gimbal-oriented  coordinate  system  (a,b,g),  where  initially 


a = Uc 


b = U 


ni 


(112) 


From  the  actual  vehicle  axes  and  the  present  desired  vehicle  axes,  attitude 
errors  can  now  be  computed. 

Consider  a three-axis  (roll,  pitch,  yaw)  maneuver  to  reorient  the 
vehicle  from  its  actual  attitude  to  the  desired  attitude,  as  three  Euler 
rotations  in  the  following  order:  roll,  pitch,  yaw.  In  matrix  form,  tins  is 


-53- 


0 


0 


B O cos  R sin  R 


cos  P 0 -sin  P I f cos  Y sin  Y 0 ~J 


-sin  Y cos  Y 0 


0 - sin  R cos  R sin  P 0 cos  P 


where  R,  P,  and  Y are  the  respective  roll,  pitch,  and  yaw  rotations. 
The  matrix  multiplication  yields 


(113) 


cos  P cos  Y 


-cos  P sm  Y 


sin  P 


sm  R sin  P cos  Y 
+ cos  R sm  Y 

sm  R sin  P sin  Y 
+ cos  R cos  Y 

-sin  R cos  P 


•cos  R sin  P cos  Y 
+ sm  R sin  Y 

cos  R sin  P sin  Y 
+ sm  R cos  Y 

cos  R cos  P 


(114) 


Simplifying  B with  small  angle  approximations  yields 


"1  Y -P 

B«  -Y  1 R 
P -R  1 


(115) 


Another  expression  for  B in  terms  of  body  axes  is 


§dm 

-r,5 

^ £dm 

• U 

n 

Ut  . 

§dm 

•U£ 

> 

B - 

0 . 
pdri) 

'°£ 

0 . 
i|dm 

• U 

n 

U . 
r)dm 

(116) 

%m 

f,dm 

• U 

n 

i 

t,dm 

•DJ 

REFERENCES 


1 E.  Da  maud , D.  L.  Kleinbub,  and  J.  B.  Shaul,  Guidance,  Control 
and  Ground  Equations  for  SSLV  C-26,  Aerospace  Corporation  Report 
No.  TOR- 01  7 2(21  { 2- 02)-  1 1 , Reissue  B (1  November  1972),  Vol.  I. 

Program  624A  Mission  Specification  for  Flight  Plan  VII- J, 

Aerospace  Corporation  Report  No.  TOR-01  72(21  1 2- 02)-6 , Rev.  1 
(28  September  1972). 

H.  Sokoloff,  Program  624A  Discrete  List  for  Flight  Plan  VII- J, 
Aerospace  Corporation  Report  No.  TOR-0172(21  12-02)-l6, 

Reissue  A (24  November  1972). 

George  W.  Cherry,  "A  General  Explicit  Optimizing  Guidance  Law 
for  Rocket  Propelled  Spacecraft,  11  Proceedings  of  the  AIAA/ION 
Astrodynamics  Guidance  and  Control  Conference,  AIAA  Paper 
No.  64-638  (24-26  August  1964). 

D.  MacPhcrson,  An  Explicit  Solution  to  the  Powered  Flight 
Dynamics  of  a Rocket  Vehicle,  Aerospace  Corporation  Report  No. 
TOR- 1 69(31  26)TN-2  (31  October  1962). 

C.  W.  Pittman,  The  Design  of  Explicit  Guidance  Equations  for 
Rocket  Ascent,  Aerospace  Corportation  Report  No. 

T DR-  469(5  5 40  - 1 0)-  4 (24  May  1965). 

F.E.  Darnaud,  A Technique  Allowing  Continuous  Operation  of 
Integral  Control,  Aerospace  Corporation  Report  No. 

TO R- 1 00 1 (2 1 1 6 -60)- 8 (1  3 September  1966). 

K.  A.  Ehricke,  "Environmental  and  Celestial  Mechanics," 

Principles  of  Guided  Missile  Design,  Vol.  I:  Space  Flight 
(D.  Van  Nostrand  Company,  Inc.,  Princeton,  N.  J.  , I960). 

Carousel  V Inertial  Navigation  System,  System  Technical  Description 
EP0137,  Delco  Electronics  Division  of  General  Motors,  Milwaukee, 
Wisconsin,  1969. 


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